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Theorem tpid2 3919
Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
tpid2.1  |-  B  e. 
_V
Assertion
Ref Expression
tpid2  |-  B  e. 
{ A ,  B ,  C }

Proof of Theorem tpid2
StepHypRef Expression
1 eqid 2437 . . 3  |-  B  =  B
213mix2i 1131 . 2  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
3 tpid2.1 . . 3  |-  B  e. 
_V
43eltp 3854 . 2  |-  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
)
52, 4mpbir 202 1  |-  B  e. 
{ A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    \/ w3o 936    = wceq 1653    e. wcel 1726   _Vcvv 2957   {ctp 3817
This theorem is referenced by:  2pthlem1  21596  2pthlem2  21597  kur14lem7  24899  brtpid2  25180  rabren3dioph  26877  el2wlkonotlem  28330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-un 3326  df-sn 3821  df-pr 3822  df-tp 3823
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